Q206 / 270 · NDA 2020 · Mathematics

Let $S=\{1,2,3, \ldots$}. A relation R on $S \times S$ is defined by $x R y$ if $\log _a x \gt \log _a y$ when $a=\frac{1}{2}$. Then the relation is

  • A
    reflexive only
  • B
    symmetric only
  • Correct answer
    C
    transitive only
  • D
    both symmetric and transitive

Solution

Give set $S=\{1,2,3, \ldots \ldots\}$ For $x R y, \log _a x \gt \log _a y$ $\Rightarrow x \gt y$ As $x R x, \log _a x \gt \log _a x$ is not valid. Hence, relation is not reflexive. For $x R y, \log _a x \gt \log _a y \Rightarrow x \gt y$ $y R x, \log _a y \gt \log _a x \Rightarrow y \gt x$ This is also not valid. Hence, relation is not symmetric also. For $x R y, \log _a x \gt \log _a y \Rightarrow x \gt y$ For $y R z, \log _a y \gt \log _a z \Rightarrow y \gt z$ So, $x R z, \log _a x \gt \log _a z \Rightarrow x \gt z$ This is a valid relation. Hence, relation is only transitive.

Hence, the correct answer is option: 'C'.

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